ECO362H1 Lecture Notes - Lecture 4: Endogenous Growth Theory, Root Mean Square, Market Power
Endogenous Growth [Optional]
1 Introduction
Objectives:
1. Extend the model to incorporate long-term endogenous growth
2. Discuss the necessary conditions for long-term growth
References:
•Romer (1990)
2 AK Model
Overview: The key idea of the AK model is that we will assume that the technology At
in period tis dependent on the average stock of capital ¯
Kin the economy. Intuitively, this
could represent that as workers use equipment, they improve on that equipment increasing
A. Specifically, we will take technology to be given by
At=A1
1−α¯
Kt(1)
where Ais a constant, which we will treat as a residual component of productivity.
As in the standard Neoclassical Growth Model, production is given by
Yt=Kα
t(AtNt)1−α
where Ktis capital employed by the representative firm. Substituting the expression for At
above gives:
Yt=AKα
t¯
K1−α
tN1−α
t
1
In equilibrium, we will have ¯
Kt=Ktand Nt= 1. Substituting these relationships into the
above productions yields:
Yt=AKt
Hence, production has an AK form as suggested by the name of the model.
It is key to note the difference between Ktand ¯
Ktand how they relate to the firms
decision making in this model. To begin, recall that we assume that there is a represen-
tative firm with production technology Kα
t(AtNt)1−αand acts as a price taker. In general,
we assume that this is because a continuum of small firms compete the prices down to the
firm’s marginal cost in equilibrium. While each of these firms consider how their choice
of Ktwill affect their profits, they do not consider how their choice of Ktwill affect the
economy through ¯
Kt. Hence, they take the path of ¯
Ktto be exogenous when solving their
optimization problem. Similarly, we assume that individual households are atomistic and do
not consider the consequences of their choice of future capital kt+1 on the economy through
¯
Kt. This effect is a spillover of the capital stock.
Household: For brevity, we will not discuss the parts of the household’s and firm’s problems
that are similar to Neoclassical Growth Model. The household’s problem is mostly identical
to the Neoclassical Growth Model:
max
{ct,xt,kt+1}∞
t=0
∞
X
t=0
βtlog(ct)(2)
subject to
ct+xt=wt+rtkt
kt+1 = (1 −δ)kt+xt
The household’s problem can then be solved similarly to the Neoclassical Growth model
and admits the same Euler Equation:
ct+1
ct
=β(1 −δ+rt+1)
Firm: As previously discussed, firms do not consider the consequences of their choice of
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capital Kton the aggregate economy. We can then write the firm’s problem as
max
Kt,Nt
Kα
t(AtNt)1−α(3)
where the interpretation is the same as before. The first-order conditions describe the firm’s
optimal behaviour:
0 = αKα−1
t(AtNt)1−α−rt
0 = (1 −α)AtKα
t(AtNt)−α−wt
Rearranging the above expressions and using (1) gives:
rt=αA Kt
¯
KtNt!α−1
wt= (1 −α)A¯
Kt Kt
¯
KtNt!α
2.1 Equilibrium
Definition 1. Given k0and technology described by (1), a competitive equilibrium is the set
of prices {wt, rt}∞
t=0; the household allocations {ct, xt, kt+1}∞
t=0; the firm allocations {Kt, Nt}∞
t=0;
such that
1. Taking prices as given, the household allocations solve the household’s problem defined
in (2);
2. Taking prices as given, the firm allocations solve the firm’s problem defined in (3);
3. Markets Clear:
•Goods Market: ct+xt=Kα
t(AtNt)1−α;
•Capital Market: Kt=kt;
•Labour Market: Nt= 1.
Other than the inclusion of technology, the definition is identical to the definition in the
Neoclassical Growth Model. We will focus on a Balanced Growth Path (BGP) equilibrium
as defined below.
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Document Summary
Objectives: extend the model to incorporate long-term endogenous growth, discuss the necessary conditions for long-term growth. Overview: the key idea of the ak model is that we will assume that the technology at in period t is dependent on the average stock of capital k in the economy. Intuitively, this could represent that as workers use equipment, they improve on that equipment increasing: speci cally, we will take technology to be given by. 1 kt (1) where a is a constant, which we will treat as a residual component of productivity. As in the standard neoclassical growth model, production is given by. Yt = k t (atnt)1 where kt is capital employed by the representative rm. In equilibrium, we will have kt = kt and nt = 1. Substituting these relationships into the above productions yields: Hence, production has an ak form as suggested by the name of the model.
